 
Summary: Probab. Th. Rel. Fields 79, 509542 (1988) Probability
TheoryRelatedFields
9 SpringerVerlag 1988
A Diffusion Limit for a Class
of RandomlyGrowing Binary Trees
David Aldous 1* and Paul Shields 2**
1Department of Statistics, University of California, BerkeleyCA 94720
2Department of Mathematics, University of Toled, ~801 W. Bancroft Street, Toledo OH 43606,
USA
Summary. Binary trees are grown by adding one node at a time, an available
node at height i being added with probability proportional to c ~, c > 1.
We establish both a "strong law of large numbers" and a "central limit
theorem" for the vector X(t)= (Xi(t)), where X~(t) is the proportion of nodes
of height i that are available at time t. We show, in fact, that there is a
deterministic process x~(t) such that
[Xi(t)xi(t)[ converges to 0 a.s.,
and such that if c > 2~,
z~(t)= 2"/2{x. +1(tc") x.+,(t~")},
and Z" (t)= (Z'~(t)), then Z"(t) converges weakly to a Gaussian diffusion Z (t).
The results are applied to establish asymptotic normality in the unbiased
